Quantum state discrimination using noisy quantum neural networks

Near-term quantum computers are noisy, and therefore must run algorithms with a low circuit depth and qubit count. Here we investigate how noise affects a quantum neural network (QNN) for state discrimination, which is applicable on near-term quantum devices as it fulfils the above criteria. We find that for the required gradient calculation on a noisy device a quantum circuit with a large number of parameters is disadvantageous. By introducing a smaller circuit ansatz we overcome this limitation, and find that the QNN performs well at noise levels of current quantum hardware. We present a model showing that the main effect of the noise is to increase the overlap between the states as circuit gates are applied, hence making discrimination more difficult. Our findings demonstrate that noisy quantum computers can be used for state discrimination and other applications, such as classifiers of the output of quantum generative adversarial networks.

Figure 1

Distribution $P\left(a\right)$ of the parameter $a$ for the input states in the first set [Eq. (1)], for an average value ${\mu }_a=0.5$ and a standard deviation ${\sigma }_a=0.15$. These values of ${\mu }_a$ and ${\sigma }_a$ are used for the data in Figs. 4–8 and 11. Due to the rather large value of ${\sigma }_a$ the most probable values of $a$ are not confined only in a thin range around ${\mu }_a$, but instead cover almost all the range between 0 and 1.

Figure 2

The general form of the quantum circuits used in this paper. The input state is on the bottom two qubits, and measuring the first qubit introduces a nonlinear dropout layer. The subcircuits $U$ and $V_{1,2}$ are shown in Fig. 3.

Figure 3

The circuits showing the trainable parameters, which are used in this paper. Comparison of results obtained for the circuits in Figs. 3 and 3 is made in Sec. 3a. (a) The $U$ and $V$ circuit blocks originally used in [10]. (b) The form of the $U$ and $V$ blocks with a reduced number of parameters.

Figure 4

The distribution of $P_{\text{inc}}$ and $P_{\text{err}}$ from 25 repeats of (a) a network biased towards reducing errors ${\alpha }_{\text{err}}=60$ and ${\alpha }_{\text{inc}}=10$ and (b) a network with a balanced cost function, ${\alpha }_{\text{err}}={\alpha }_{\text{inc}}=40$, both with values ${\mu }_a=0.5$ and ${\sigma }_a=0.15$. An example undesirable output for a single minimizing error run is in the inset, where no $b$ states are measured correctly, but the network still converges (the $x$ axis shows the output label and the color is the input state). The interquartile range is contained within the box, and the 5th and 95th percentiles are marked by the whiskers. Outliers of this range are marked by a diamond. The mean is marked with a white square, and the median is the line across the box.

Figure 5

The distributions of loss ($P_{\text{err}}+P_{\text{inc}}$) at different noise levels for the two circuits shown in Fig. 3. Both have other parameters fixed, ${\mu }_a=0.5,{\sigma }_a=0.15,{\alpha }_{\text{err}}={\alpha }_{\text{inc}}=40$. We observe that reducing the number of parameters is advantageous at all noise levels.

Figure 6

Evolution of the normalized cost functions for larger and reduced circuits for ${\mu }_a=0.5$ and ${\sigma }_a=0.15$, with noise levels of 0.001 and 0.1. Shown here is the number of steps taken to converge. Note that the time taken to complete a single step of the longer circuit is much greater than for the reduced circuit.

Figure 7

The distribution of loss for 25 repeats of training the network. The cost function is balanced, ${\alpha }_{\text{err}}={\alpha }_{\text{inc}}=40$, ${\mu }_a=0.5$, and ${\sigma }_a=0.15$. At levels of noise present in current devices, 0.01, the loss value is favorable, an average of 0.2.

Figure 8

Distribution of loss ($P_{\text{err}}+P_{\text{inc}}$) against training noise for different noise levels in the validation circuit: (a) 0.0, (b) 0.001, (c) 0.01, and (d) 0.05.

Figure 9

The distribution of loss ($P_{\text{err}}+P_{\text{inc}}$) and the effect of different values of ${\mu }_a$. The cost function is balanced, ${\alpha }_{\text{err}}={\alpha }_{\text{inc}}=40$, and ${\sigma }_a=0.15$. The noise level is (a) 0.0, (b) 0.001, (c) 0.01, and (d) 0.05. We see that for lower values of ${\mu }_a$, corresponding to smaller overlap between the states to be discriminated, the discrimination task is performed better. The red stars indicate the fidelity $F_{\tilde{a}\tilde{b}}$ between the two states after three applications of Kraus operators to each of the data qubits, as given by Eq. (14d).

Figure 10

Fidelities as function of the number of applied noise channels, $n$, (a) between the same states with noise applied to one state ($F_{a\tilde{a}}$), (b) between the two different states with noise applied to one state ($F_{a\tilde{b}}=F_{b\tilde{a}}$), and (c) with noise applied to both states ($F_{\tilde{a}\tilde{b}}$). Markers show the calculated numeric fidelities using Eq. (13), and lines show the low-order expansions given by Eq. (14). The low-order expansion agrees well with the numerical results for all cases.

Figure 11

The distribution of loss and ${\theta }_{10}$ obtained at different noise levels. Here we can see the effect of noise on the values of ${\theta }_{10}$ that the optimizer converges to. We only show a single representative parameter ${\theta }_{10}$, since we have an approximately similar behavior for all other parameters. This is shown at different noise levels: (a) 0.0, (b) 0.001, (c) 0.01, and (d) 0.05.